I am going to give something of a roundabout answer to the question to explain the pronunciation of this name.

Montaigne is an older spelling of montagne. Although today the pronunciation of the name is almost always [mɔ̃tɛɲ], in his time it was pronounced [mɔ̃taɲ]. The new pronunciation is in fact a spelling pronunciation (as when some people pronounce the th in the English word clothes).

At one time the sound now spelt gn was very often spelt ign. For example, cogner was coigner, gagner was gaigner. The spelling reform that replaced almost all occurrences of ign with gn was for some reason not applied to certain words. The word oignon, for example, has retained its original spelling, but unlike Montaigne, also its original pronunciation [ɔɲɔ̃]. On the other hand, the word poigne has undergone the same process as Montaigne. It was originally pronounced [pɔɲ], but since the meaning of the i in the spelling is no longer understood by French-speakers, most people now pronounce this word [pwaɲ].

like every where, a name gives the quality/the dignity to a man : he is his name ; it cannot be translated. but, if you wish it, you can understand “montaîgne” like a owner of a forest which the fruit is “chataîgnes” or “Fougères” – “mon” means mine and “aîgne” my proptriety.